Theorem tfinds 3212

Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197.

Hypotheses

Ref	Expression
tfinds.1	|- (x = (/) -> (ph <-> ps))
tfinds.2	|- (x = y -> (ph <-> ch))
tfinds.3	|- (x = suc y -> (ph <-> th))
tfinds.4	|- (x = A -> (ph <-> ta))
tfinds.5	|- ps
tfinds.6	|- (y e. On -> (ch -> th))
tfinds.7	|- (Lim x -> (A.y e. x ch -> ph))

Assertion

Ref	Expression
tfinds	|- (A e. On -> ta)

Distinct variable groups:   x,y   x,A   ch,x   ta,x   ph,y

Proof of Theorem tfinds

Step	Hyp	Ref	Expression
1		tfinds.2	. 2 |- (x = y -> (ph <-> ch))
2		tfinds.4	. 2 |- (x = A -> (ph <-> ta))
3		eloni 2985	. . . . 5 |- (x e. On -> Ord x)
4		df-lim 2980	. . . . . . . . . . . . . . . 16 |- (Lim x <-> (Ord x /\ x =/= (/) /\ x = U.x))
5	4	biimpri 150	. . . . . . . . . . . . . . 15 |- ((Ord x /\ x =/= (/) /\ x = U.x) -> Lim x)
6	5	3com23 845	. . . . . . . . . . . . . 14 |- ((Ord x /\ x = U.x /\ x =/= (/)) -> Lim x)
7	6	3expia 841	. . . . . . . . . . . . 13 |- ((Ord x /\ x = U.x) -> (x =/= (/) -> Lim x))
8	7	necon1bd 1676	. . . . . . . . . . . 12 |- ((Ord x /\ x = U.x) -> (-. Lim x -> x = (/)))
9	8	ex 371	. . . . . . . . . . 11 |- (Ord x -> (x = U.x -> (-. Lim x -> x = (/))))
10	9	com23 32	. . . . . . . . . 10 |- (Ord x -> (-. Lim x -> (x = U.x -> x = (/))))
11		orduninsuc 3197	. . . . . . . . . . 11 |- (Ord x -> (x = U.x <-> -. E.y e. On x = suc y))
12	11	biimprd 152	. . . . . . . . . 10 |- (Ord x -> (-. E.y e. On x = suc y -> x = U.x))
13	10, 12	syl5d 55	. . . . . . . . 9 |- (Ord x -> (-. Lim x -> (-. E.y e. On x = suc y -> x = (/))))
14	13	imp 348	. . . . . . . 8 |- ((Ord x /\ -. Lim x) -> (-. E.y e. On x = suc y -> x = (/)))
15	14	con1d 93	. . . . . . 7 |- ((Ord x /\ -. Lim x) -> (-. x = (/) -> E.y e. On x = suc y))
16	15	orrd 231	. . . . . 6 |- ((Ord x /\ -. Lim x) -> (x = (/) \/ E.y e. On x = suc y))
17	16	ex 371	. . . . 5 |- (Ord x -> (-. Lim x -> (x = (/) \/ E.y e. On x = suc y)))
18	3, 17	syl 10	. . . 4 |- (x e. On -> (-. Lim x -> (x = (/) \/ E.y e. On x = suc y)))
19		tfinds.5	. . . . . . 7 |- ps
20		tfinds.1	. . . . . . 7 |- (x = (/) -> (ph <-> ps))
21	19, 20	mpbiri 192	. . . . . 6 |- (x = (/) -> ph)
22	21	a1d 12	. . . . 5 |- (x = (/) -> (A.y e. x ch -> ph))
23		hbra1 1733	. . . . . . 7 |- (A.y e. x ch -> A.yA.y e. x ch)
24		ax-17 1007	. . . . . . 7 |- (ph -> A.yph)
25	23, 24	hbim 1043	. . . . . 6 |- ((A.y e. x ch -> ph) -> A.y(A.y e. x ch -> ph))
26		raleq1 1832	. . . . . . . . . . 11 |- (x = suc y -> (A.z e. x [z / x]ph <-> A.z e. suc y[z / x]ph))
27		sbequ 1266	. . . . . . . . . . . . 13 |- (y = z -> ([y / x]ph <-> [z / x]ph))
28		ax-17 1007	. . . . . . . . . . . . . 14 |- (ch -> A.xch)
29	28, 1	sbie 1233	. . . . . . . . . . . . 13 |- ([y / x]ph <-> ch)
30	27, 29	syl5bbr 537	. . . . . . . . . . . 12 |- (y = z -> (ch <-> [z / x]ph))
31	30	cbvralv 1846	. . . . . . . . . . 11 |- (A.y e. x ch <-> A.z e. x [z / x]ph)
32		ax-17 1007	. . . . . . . . . . . 12 |- (ph -> A.zph)
33		hbs1 1371	. . . . . . . . . . . 12 |- ([z / x]ph -> A.x[z / x]ph)
34		sbequ12 1218	. . . . . . . . . . . 12 |- (x = z -> (ph <-> [z / x]ph))
35	32, 33, 34	cbvral 1844	. . . . . . . . . . 11 |- (A.x e. suc yph <-> A.z e. suc y[z / x]ph)
36	26, 31, 35	3bitr4g 558	. . . . . . . . . 10 |- (x = suc y -> (A.y e. x ch <-> A.x e. suc yph))
37	36	biimpd 151	. . . . . . . . 9 |- (x = suc y -> (A.y e. x ch -> A.x e. suc yph))
38		tfinds.6	. . . . . . . . . 10 |- (y e. On -> (ch -> th))
39		visset 1859	. . . . . . . . . . . 12 |- y e. V
40	39	sucid 3051	. . . . . . . . . . 11 |- y e. suc y
41	1	rcla4v 1919	. . . . . . . . . . 11 |- (y e. suc y -> (A.x e. suc yph -> ch))
42	40, 41	ax-mp 7	. . . . . . . . . 10 |- (A.x e. suc yph -> ch)
43	38, 42	syl5 21	. . . . . . . . 9 |- (y e. On -> (A.x e. suc yph -> th))
44	37, 43	sylan9r 471	. . . . . . . 8 |- ((y e. On /\ x = suc y) -> (A.y e. x ch -> th))
45		tfinds.3	. . . . . . . . 9 |- (x = suc y -> (ph <-> th))
46	45	adantl 388	. . . . . . . 8 |- ((y e. On /\ x = suc y) -> (ph <-> th))
47	44, 46	sylibrd 202	. . . . . . 7 |- ((y e. On /\ x = suc y) -> (A.y e. x ch -> ph))
48	47	ex 371	. . . . . 6 |- (y e. On -> (x = suc y -> (A.y e. x ch -> ph)))
49	25, 48	r19.23ai 1788	. . . . 5 |- (E.y e. On x = suc y -> (A.y e. x ch -> ph))
50	22, 49	jaoi 339	. . . 4 |- ((x = (/) \/ E.y e. On x = suc y) -> (A.y e. x ch -> ph))
51	18, 50	syl6 22	. . 3 |- (x e. On -> (-. Lim x -> (A.y e. x ch -> ph)))
52		tfinds.7	. . 3 |- (Lim x -> (A.y e. x ch -> ph))
53	51, 52	pm2.61d2 127	. 2 |- (x e. On -> (A.y e. x ch -> ph))
54	1, 2, 53	tfis3 3211	1 |- (A e. On -> ta)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   /\ w3a 781   = wceq 992   e. wcel 994  [wsbc 1207   =/= wne 1628  A.wral 1691  E.wrex 1692  (/)c0 2332  U.cuni 2569  Ord word 2974  Oncon0 2975  Lim wlim 2976  suc csuc 2977

This theorem is referenced by:  tfindsg 3213  tfindes 3215  tfinds3 3217  oa0r 4309  om0r 4310  om1r 4313  oe1m 4315  oeoalem 4359  r1tr 4800  alephon 5015  alephcard 5017  alephordi 5024  elomsubsd 11446  omsublim 11448

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-sbc 1987  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981

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